





"""simple docstring"""



import inspect
import logging
import os
import random
import shutil
import tempfile
import unittest

import pytest
import torch
from torch import nn
from torch.utils.data import DataLoader, TensorDataset

from accelerate import Accelerator
from accelerate.test_utils import execute_subprocess_async, require_cuda
from accelerate.utils import ProjectConfiguration, set_seed


a_					=     logging.getLogger(__name__)







def 						__lowercase	(  snake_case_    :    List[Any]=2							,snake_case_    :    Tuple=3							,snake_case_    :    Optional[Any]=16							,snake_case_    :    Optional[int] = 10							,snake_case_    :    Union[str, Any] = 2		)  ->Optional[Any]:



    '''simple docstring'''







    def get_dataset(snake_case_    :    int		):
        __A       :		Dict							=	torch.randn(batch_size * n_batches							,1		)
        return TensorDataset(__lowerCAmelCase							,a * x + b + 0.1 * torch.randn(batch_size * n_batches							,1		)		)

    __A       :		Union[str, Any]							=	get_dataset(__lowerCAmelCase		)
    __A       :		Union[str, Any]							=	get_dataset(__lowerCAmelCase		)
    __A       :		Optional[int]							=	DataLoader(__lowerCAmelCase							,shuffle=__lowerCAmelCase							,batch_size=__lowerCAmelCase							,num_workers=4		)
    __A       :		Union[str, Any]							=	DataLoader(__lowerCAmelCase							,shuffle=__lowerCAmelCase							,batch_size=__lowerCAmelCase							,num_workers=4		)
    return (train_dataloader, valid_dataloader)







def 						__lowercase	(  snake_case_    :    Optional[int]							,snake_case_    :    List[str]							,snake_case_    :    str							,snake_case_    :    Union[str, Any]							,snake_case_    :    Any							,snake_case_    :    Optional[int]=None		)  ->Union[str, Any]:



    '''simple docstring'''







    __A       :		List[Any]							=	[]
    for epoch in range(__lowerCAmelCase		):
        # Train quickly
        model.train()
        for batch in dataloader:
            __A					,			__A       :		Dict							=	batch
            __A       :		Tuple							=	model(__lowerCAmelCase		)
            __A       :		str							=	torch.nn.functional.mse_loss(__lowerCAmelCase							,__lowerCAmelCase		)
            accelerator.backward(__lowerCAmelCase		)
            optimizer.step()
            optimizer.zero_grad()
        rands.append(random.random()		)  # Introduce some randomness
        if scheduler is not None:
            scheduler.step()
    return rands






class 						__snake_case   (						nn.Module	):





 """simple docstring"""


 def __init__(       self    ):



     '''simple docstring'''




     super().__init__()
     __A       :		int							=	nn.Parameter(torch.randn(1    )    )
     __A       :		Dict							=	nn.Parameter(torch.randn(1    )    )







 def    UpperCamelCase__(       self   ,		__lowerCamelCase    ):



     '''simple docstring'''




     return x * self.a + self.b






class 						__snake_case   (						unittest.TestCase	):





 """simple docstring"""


 def    UpperCamelCase__(       self    ):



     '''simple docstring'''




     with tempfile.TemporaryDirectory() as tmpdir:
         set_seed(42    )
         __A       :		List[Any]							=	DummyModel()
         __A       :		Any							=	torch.optim.Adam(params=model.parameters()   ,		lr=1e-3    )
         __A					,			__A       :		Union[str, Any]							=	dummy_dataloaders()
         __A       :		Optional[Any]							=	ProjectConfiguration(total_limit=1   ,		project_dir=_lowerCamelCase   ,		automatic_checkpoint_naming=_lowerCamelCase    )
         # Train baseline
         __A       :		List[str]							=	Accelerator(project_config=_lowerCamelCase    )
         __A					,			__A					,			__A					,			__A       :		Optional[Any]							=	accelerator.prepare(
             _lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         # Save initial
         accelerator.save_state()

         # Save second state
         accelerator.save_state()
         self.assertEqual(len(os.listdir(accelerator.project_dir    )    )   ,		1    )

 def    UpperCamelCase__(       self    ):



     '''simple docstring'''




     with tempfile.TemporaryDirectory() as tmpdir:
         set_seed(42    )
         __A       :		Tuple							=	DummyModel()
         __A       :		Tuple							=	torch.optim.Adam(params=model.parameters()   ,		lr=1e-3    )
         __A					,			__A       :		List[str]							=	dummy_dataloaders()
         # Train baseline
         __A       :		str							=	Accelerator()
         __A					,			__A					,			__A					,			__A       :		str							=	accelerator.prepare(
             _lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         # Save initial
         __A       :		Optional[int]							=	os.path.join(_lowerCamelCase   ,		'''initial'''    )
         accelerator.save_state(_lowerCamelCase    )
         ((__A)					,			(__A))       :		Tuple							=	model.a.item(), model.b.item()
         __A       :		Union[str, Any]							=	optimizer.state_dict()
         __A       :		str							=	train(3   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         ((__A)					,			(__A))       :		str							=	model.a.item(), model.b.item()
         __A       :		List[Any]							=	optimizer.state_dict()

         # Train partially
         set_seed(42    )
         __A       :		List[str]							=	DummyModel()
         __A       :		List[Any]							=	torch.optim.Adam(params=model.parameters()   ,		lr=1e-3    )
         __A					,			__A       :		List[str]							=	dummy_dataloaders()
         __A       :		Tuple							=	Accelerator()
         __A					,			__A					,			__A					,			__A       :		int							=	accelerator.prepare(
             _lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         accelerator.load_state(_lowerCamelCase    )
         ((__A)					,			(__A))       :		List[str]							=	model.a.item(), model.b.item()
         __A       :		List[str]							=	optimizer.state_dict()
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )

         __A       :		Optional[int]							=	train(2   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         # Save everything
         __A       :		int							=	os.path.join(_lowerCamelCase   ,		'''checkpoint'''    )
         accelerator.save_state(_lowerCamelCase    )

         # Load everything back in and make sure all states work
         accelerator.load_state(_lowerCamelCase    )
         test_rands += train(1   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         ((__A)					,			(__A))       :		str							=	model.a.item(), model.b.item()
         __A       :		List[str]							=	optimizer.state_dict()
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )

 def    UpperCamelCase__(       self    ):



     '''simple docstring'''




     with tempfile.TemporaryDirectory() as tmpdir:
         set_seed(42    )
         __A       :		str							=	DummyModel()
         __A       :		List[Any]							=	torch.optim.Adam(params=model.parameters()   ,		lr=1e-3    )
         __A					,			__A       :		Any							=	dummy_dataloaders()
         __A       :		Union[str, Any]							=	ProjectConfiguration(automatic_checkpoint_naming=_lowerCamelCase    )

         # Train baseline
         __A       :		List[Any]							=	Accelerator(project_dir=_lowerCamelCase   ,		project_config=_lowerCamelCase    )
         __A					,			__A					,			__A					,			__A       :		Dict							=	accelerator.prepare(
             _lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         # Save initial
         accelerator.save_state()
         ((__A)					,			(__A))       :		Union[str, Any]							=	model.a.item(), model.b.item()
         __A       :		Union[str, Any]							=	optimizer.state_dict()
         __A       :		Tuple							=	train(3   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         ((__A)					,			(__A))       :		str							=	model.a.item(), model.b.item()
         __A       :		Tuple							=	optimizer.state_dict()

         # Train partially
         set_seed(42    )
         __A       :		int							=	DummyModel()
         __A       :		Union[str, Any]							=	torch.optim.Adam(params=model.parameters()   ,		lr=1e-3    )
         __A					,			__A       :		Any							=	dummy_dataloaders()
         __A       :		Union[str, Any]							=	ProjectConfiguration(iteration=1   ,		automatic_checkpoint_naming=_lowerCamelCase    )
         __A       :		Union[str, Any]							=	Accelerator(project_dir=_lowerCamelCase   ,		project_config=_lowerCamelCase    )
         __A					,			__A					,			__A					,			__A       :		Tuple							=	accelerator.prepare(
             _lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         accelerator.load_state(os.path.join(_lowerCamelCase   ,		'''checkpoints'''   ,		'''checkpoint_0'''    )    )
         ((__A)					,			(__A))       :		Union[str, Any]							=	model.a.item(), model.b.item()
         __A       :		Optional[Any]							=	optimizer.state_dict()
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )

         __A       :		Tuple							=	train(2   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         # Save everything
         accelerator.save_state()

         # Load everything back in and make sure all states work
         accelerator.load_state(os.path.join(_lowerCamelCase   ,		'''checkpoints'''   ,		'''checkpoint_1'''    )    )
         test_rands += train(1   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         ((__A)					,			(__A))       :		List[str]							=	model.a.item(), model.b.item()
         __A       :		List[str]							=	optimizer.state_dict()
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertEqual(_lowerCamelCase   ,		_lowerCamelCase    )

 def    UpperCamelCase__(       self    ):



     '''simple docstring'''




     __A       :		Tuple							=	torch.tensor([1, 2, 3]    )
     __A       :		int							=	torch.tensor([2, 3, 4]    )
     __A       :		Dict							=	DummyModel()
     __A       :		Optional[Any]							=	torch.optim.Adam(net.parameters()    )
     __A       :		Tuple							=	Accelerator()
     with self.assertRaises(_lowerCamelCase    ) as ve:
         accelerator.register_for_checkpointing(_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
     __A       :		Union[str, Any]							=	str(ve.exception    )
     self.assertTrue('''Item at index 0''' in message    )
     self.assertTrue('''Item at index 1''' in message    )
     self.assertFalse('''Item at index 2''' in message    )
     self.assertFalse('''Item at index 3''' in message    )

 def    UpperCamelCase__(       self    ):



     '''simple docstring'''




     with tempfile.TemporaryDirectory() as tmpdir:
         set_seed(42    )
         __A       :		int							=	DummyModel()
         __A       :		Union[str, Any]							=	torch.optim.Adam(params=model.parameters()   ,		lr=1e-3    )
         __A       :		Union[str, Any]							=	torch.optim.lr_scheduler.StepLR(_lowerCamelCase   ,		step_size=1   ,		gamma=0.9_9    )
         __A					,			__A       :		int							=	dummy_dataloaders()
         __A       :		str							=	ProjectConfiguration(automatic_checkpoint_naming=_lowerCamelCase    )
         # Train baseline
         __A       :		Optional[int]							=	Accelerator(project_dir=_lowerCamelCase   ,		project_config=_lowerCamelCase    )
         __A					,			__A					,			__A					,			__A					,			__A       :		Union[str, Any]							=	accelerator.prepare(
             _lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         # Save initial
         accelerator.save_state()
         __A       :		List[Any]							=	scheduler.state_dict()
         train(3   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase   ,		_lowerCamelCase    )
         self.assertNotEqual(_lowerCamelCase   ,		scheduler.state_dict()    )

         # Load everything back in and make sure all states work
         accelerator.load_state(os.path.join(_lowerCamelCase   ,		'''checkpoints'''   ,		'''checkpoint_0'''    )    )
         self.assertEqual(_lowerCamelCase   ,		scheduler.state_dict()    )

 def    UpperCamelCase__(       self    ):



     '''simple docstring'''




     with tempfile.TemporaryDirectory() as tmpdir:
         set_seed(42    )
         __A       :		Any							=	DummyModel()
         __A       :		List[Any]							=	ProjectConfiguration(automatic_checkpoint_naming=_lowerCamelCase   ,		total_limit=2    )
         # Train baseline
         __A       :		List[Any]							=	Accelerator(project_dir=_lowerCamelCase   ,		project_config=_lowerCamelCase    )
         __A       :		Optional[int]							=	accelerator.prepare(_lowerCamelCase    )
         # Save 3 states:
         for _ in range(11    ):
             accelerator.save_state()
         self.assertTrue(not os.path.exists(os.path.join(_lowerCamelCase   ,		'''checkpoints'''   ,		'''checkpoint_0'''    )    )    )
         self.assertTrue(os.path.exists(os.path.join(_lowerCamelCase   ,		'''checkpoints'''   ,		'''checkpoint_9'''    )    )    )
         self.assertTrue(os.path.exists(os.path.join(_lowerCamelCase   ,		'''checkpoints'''   ,		'''checkpoint_10'''    )    )    )







 @require_cuda
 def    UpperCamelCase__(       self    ):



     '''simple docstring'''




     __A       :		Union[str, Any]							=	['''torchrun''', F"""--nproc_per_node={torch.cuda.device_count()}""", inspect.getfile(self.__class__    )]
     execute_subprocess_async(_lowerCamelCase   ,		env=os.environ.copy()    )


if __name__ == "__main__":
       a_					=     """/tmp/accelerate/state_checkpointing"""
       a_					=     DummyModel()
       a_					=     torch.optim.Adam(params=model.parameters(), lr=1e-3)
       a_					=     torch.optim.lr_scheduler.StepLR(optimizer, step_size=1, gamma=0.99)
       a_					,     a_					=     dummy_dataloaders()
       a_					=     ProjectConfiguration(automatic_checkpoint_naming=True)
       # Train baseline
       a_					=     Accelerator(project_dir=savedir, project_config=project_config, mixed_precision="""no""")
       if accelerator.process_index == 0:
              if os.path.exists(savedir):
                     shutil.rmtree(savedir)
              os.makedirs(savedir)
       a_					,     a_					,     a_					,     a_					,     a_					=     accelerator.prepare(
           model, optimizer, train_dataloader, valid_dataloader, scheduler
       )
       a_					,     a_					=     accelerator.prepare(model, optimizer)
       train(3, model, train_dataloader, optimizer, accelerator, scheduler)
       # Check that the intial optimizer is loaded on the GPU
       for group in optimizer.param_groups:
              a_					=     group["""params"""][0].device
              break
       assert param_device.type == accelerator.device.type
       a_					=     model.cpu()
       accelerator.wait_for_everyone()
       accelerator.save_state()
       accelerator.wait_for_everyone()

       # Check CPU state
       accelerator.load_state(os.path.join(savedir, """checkpoints""", """checkpoint_0"""), map_location="""cpu""")
       for group in optimizer.param_groups:
              a_					=     group["""params"""][0].device
              break
       assert (
           param_device.type == torch.device("""cpu""").type
       ), f"Loaded optimizer states did not match, expected to be loaded on the CPU but got {param_device}"

       # Check device state
       model.to(accelerator.device)
       accelerator.load_state(os.path.join(savedir, """checkpoints""", """checkpoint_0"""), map_location="""on_device""")
       for group in optimizer.param_groups:
              a_					=     group["""params"""][0].device
              break
       assert (
           param_device.type == accelerator.device.type
       ), f"Loaded optimizer states did not match, expected to be loaded on {accelerator.device} but got {param_device}"

       # Check error
       with pytest.raises(TypeError, match="""Unsupported optimizer map location passed"""):
              accelerator.load_state(os.path.join(savedir, """checkpoints""", """checkpoint_0"""), map_location="""invalid""")
       accelerator.wait_for_everyone()
       if accelerator.process_index == 0:
              shutil.rmtree(savedir)
       accelerator.wait_for_everyone()





"""simple docstring"""


from math import sqrt





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   bool:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number >= 0
						), "'number' must been an int and positive"

						lowerCamelCase__   =True

						# 0 and 1 are none primes.
						if number <= 1:
												lowerCamelCase__   =False

						for divisor in range(2  ,       int(round(sqrt(__lowerCAmelCase							)							)							) + 1							):
												# if 'number' divisible by 'divisor' then sets 'status'
												# of false and break up the loop.
												if number % divisor == 0:
																		lowerCamelCase__   =False
																		break

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'status' must been from type bool"

						return status





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[str]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n > 2), "'N' must been an int and > 2"

						# beginList: contains all natural numbers from 2 up to N
						lowerCamelCase__   =list(range(2  ,       n + 1							)							)

						lowerCamelCase__   =[]  # this list will be returns.

						# actual sieve of erathostenes
						for i in range(len(__lowerCAmelCase							)							):
												for j in range(i + 1  ,       len(__lowerCAmelCase							)							):
																		if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
																								lowerCamelCase__   =0

    # filters actual prime numbers.
						lowerCamelCase__   =[x for x in begin_list if x != 0]

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type list"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Optional[int]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n > 2), "'N' must been an int and > 2"

						lowerCamelCase__   =[]

						# iterates over all numbers between 2 up to N+1
						# if a number is prime then appends to list 'ans'
						for number in range(2  ,       n + 1							):
												if is_prime(__lowerCAmelCase							):
																		ans.append(__lowerCAmelCase							)

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type list"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and number >= 0, "'number' must been an int and >= 0"

						lowerCamelCase__   =[]  # this list will be returns of the function.

						# potential prime number factors.

						lowerCamelCase__   =2

						lowerCamelCase__   =number

						if number == 0 or number == 1:
												ans.append(__lowerCAmelCase							)

						# if 'number' not prime then builds the prime factorization of 'number'
						elif not is_prime(__lowerCAmelCase							):
												while quotient != 1:
																		if is_prime(__lowerCAmelCase							) and (quotient % factor == 0):
																								ans.append(__lowerCAmelCase							)
																								quotient /= factor
																		else:
																								factor += 1

						else:
												ans.append(__lowerCAmelCase							)

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type list"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Optional[int]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						lowerCamelCase__   =0

						# prime factorization of 'number'
						lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)

						lowerCamelCase__   =max(__lowerCAmelCase							)

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type int"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						lowerCamelCase__   =0

						# prime factorization of 'number'
						lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)

						lowerCamelCase__   =min(__lowerCAmelCase							)

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type int"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'number' must been an int"
						assert isinstance(number % 2 == 0  ,       __lowerCAmelCase							), "compare bust been from type bool"

						return number % 2 == 0





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'number' must been an int"
						assert isinstance(number % 2 != 0  ,       __lowerCAmelCase							), "compare bust been from type bool"

						return number % 2 != 0





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   int:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (number > 2) and is_even(__lowerCAmelCase							)
						), "'number' must been an int, even and > 2"

						lowerCamelCase__   =[]  # this list will returned

						# creates a list of prime numbers between 2 up to 'number'
						lowerCamelCase__   =get_prime_numbers(__lowerCAmelCase							)
						lowerCamelCase__   =len(__lowerCAmelCase							)

						# run variable for while-loops.
						lowerCamelCase__   =0
						lowerCamelCase__   =None

						# exit variable. for break up the loops
						lowerCamelCase__   =True

						while i < len_pn and loop:
												lowerCamelCase__   =i + 1

												while j < len_pn and loop:
																		if prime_numbers[i] + prime_numbers[j] == number:
																								lowerCamelCase__   =False
																								ans.append(prime_numbers[i]							)
																								ans.append(prime_numbers[j]							)

																		j += 1

												i += 1

						# precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (len(__lowerCAmelCase							) == 2)
						    and (ans[0] + ans[1] == number)
						    and is_prime(ans[0]							)
						    and is_prime(ans[1]							)
						), "'ans' must contains two primes. And sum of elements must been eq 'number'"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   Tuple:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (numbera >= 0)
						    and (numbera >= 0)
						), "'number1' and 'number2' must been positive integer."

						lowerCamelCase__   =0

						while numbera != 0:
												lowerCamelCase__   =numbera % numbera
												lowerCamelCase__   =numbera
												lowerCamelCase__   =rest

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    numbera >= 0
						), "'number' must been from type int and positive"

						return numbera





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (numbera >= 1)
						    and (numbera >= 1)
						), "'number1' and 'number2' must been positive integer."

						lowerCamelCase__   =1  # actual answer that will be return.

						# for kgV (x,1)
						if numbera > 1 and numbera > 1:
												# builds the prime factorization of 'number1' and 'number2'
												lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)
												lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)

						elif numbera == 1 or numbera == 1:
												lowerCamelCase__   =[]
												lowerCamelCase__   =[]
												lowerCamelCase__   =max(__lowerCAmelCase  ,       __lowerCAmelCase							)

						lowerCamelCase__   =0
						lowerCamelCase__   =0

						lowerCamelCase__   =[]  # captured numbers int both 'primeFac1' and 'primeFac2'

						# iterates through primeFac1
						for n in prime_fac_a:
												if n not in done:
																		if n in prime_fac_a:
																								lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)
																								lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)

																								for _ in range(max(__lowerCAmelCase  ,       __lowerCAmelCase							)							):
																														ans *= n

																		else:
																								lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)

																								for _ in range(__lowerCAmelCase							):
																														ans *= n

																		done.append(__lowerCAmelCase							)

    # iterates through primeFac2
						for n in prime_fac_a:
												if n not in done:
																		lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)

																		for _ in range(__lowerCAmelCase							):
																								ans *= n

																		done.append(__lowerCAmelCase							)

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    ans >= 0
						), "'ans' must been from type int and positive"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Optional[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 0), "'number' must been a positive int"

						lowerCamelCase__   =0
						lowerCamelCase__   =2  # this variable holds the answer

						while index < n:
												index += 1

												ans += 1  # counts to the next number

												# if ans not prime then
												# runs to the next prime number.
												while not is_prime(__lowerCAmelCase							):
																		ans += 1

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and is_prime(
						    __lowerCAmelCase							), "'ans' must been a prime number and from type int"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   int:




						'''simple docstring'''



						assert (
						    is_prime(__lowerCAmelCase							) and is_prime(__lowerCAmelCase							) and (p_number_a < p_number_a)
						), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

						lowerCamelCase__   =p_number_a + 1  # jump to the next number

						lowerCamelCase__   =[]  # this list will be returns.

						# if number is not prime then
						# fetch the next prime number.
						while not is_prime(__lowerCAmelCase							):
												number += 1

						while number < p_number_a:
												ans.append(__lowerCAmelCase							)

												number += 1

												# fetch the next prime number.
												while not is_prime(__lowerCAmelCase							):
																		number += 1

    # precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and ans[0] != p_number_a
						    and ans[len(__lowerCAmelCase							) - 1] != p_number_a
						), "'ans' must been a list without the arguments"

						# 'ans' contains not 'pNumber1' and 'pNumber2' !
						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   int:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 1), "'n' must been int and >= 1"

						lowerCamelCase__   =[]  # will be returned.

						for divisor in range(1  ,       n + 1							):
												if n % divisor == 0:
																		ans.append(__lowerCAmelCase							)

    # precondition
						assert ans[0] == 1 and ans[len(__lowerCAmelCase							) - 1] == n, "Error in function getDivisiors(...)"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number > 1
						), "'number' must been an int and >= 1"

						lowerCamelCase__   =get_divisors(__lowerCAmelCase							)

						# precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (divisors[0] == 1)
						    and (divisors[len(__lowerCAmelCase							) - 1] == number)
						), "Error in help-function getDivisiors(...)"

						# summed all divisors up to 'number' (exclusive), hence [:-1]
						return sum(divisors[:-1]							) == number





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   Dict:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (denominator != 0)
						), "The arguments must been from type int and 'denominator' != 0"

						# build the greatest common divisor of numerator and denominator.
						lowerCamelCase__   =gcd(abs(__lowerCAmelCase							)  ,       abs(__lowerCAmelCase							)							)

						# precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (numerator % gcd_of_fraction == 0)
						    and (denominator % gcd_of_fraction == 0)
						), "Error in function gcd(...,...)"

						return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 0), "'n' must been a int and >= 0"

						lowerCamelCase__   =1  # this will be return.

						for factor in range(1  ,       n + 1							):
												ans *= factor

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Tuple:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 0), "'n' must been an int and >= 0"

						lowerCamelCase__   =0
						lowerCamelCase__   =1
						lowerCamelCase__   =1  # this will be return

						for _ in range(n - 1							):
												lowerCamelCase__   =ans
												ans += fiba
												lowerCamelCase__   =tmp

						return ans


